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FACULTY WORKING
PAPER NO. 865
Every Finite Distributive Lattice
Stable Matchings
Is
a Set cf
Charles Blair
<*>i£
Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois. Urbana-Chanpaign
Co-lege of
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BEBR
FACULTY WORKING PAPER NO. 865
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
April 1982
Every Finite Distributive Lattice
Is a Set of Stable Matchings
Charles Blair, Associate Professor
Department of Business Administration
Abstract
We show that, given a lattice,
a set of men and women with
preferences can be constructed whose
stable matchings are precisely
that
lattice.
This is a converse of a result
of J. H. Conway.
Every Finite Distributive Lattice is a Set of Stable Matchings
by
Charles Blair
Suppose we have n men and n women.
Each of the 2n people has a
linear preference ordering on those of the opposite sex.
ested in matchings to form n couples.
We are inter-
A matching is stable if we cannot
find a woman in one couple and a man in another who would prefer each
other to their present partners.
Stable matchings were first defined by Gale and Shapley [1], who
showed that for any preference orderings a stable matching always exists.
In general, there will be several stable matchings.
For example, if
all the men happen to have different first preferences, giving each man
his first choice will be stable, regardless of the women's preferences.
Similarly, giving each woman her first choice (if possible) will be
stable.
Conway [2, p. 87-92] defines a partial ordering on the set of stable
matchings as follows:
one matching is
>_
another if every man is at least
as happy with his partner in the first matching as with his partner in
the second.
He shows that the set of stable matchings is always a finite
distributive lattice.
Knuth 12, p. 92] asks whether every finite dis-
tributive lattice can occur as the set of stable matchings generated by
some set of men and women.
We show this is the case.
We will require some preliminary facts about lattices.
distributive lattice and x € L let V = {v
L
|y <_x} be
If L is a
disjoint from L.
is the partial ordering on L u V defined by (i) if w,z e L then
w
>_
z
-2-
x
iff w
in L
>
—
w €
(ii) if
in L.
z
L,
v
e V w
z
>
—
v
z
iff w
z in L.
>
—
(iii)
•yr
> v
iff w >
—
w — z
v
w
for any w,z.
is a distributive lattice,
v
z»
Intuitively, L
is formed from L by making a copy of all the elements
(iv)
f
—
z
L
x
<_
and putting the copies immediately below the originals.
If a set S of lattices includes a one-element lattice and
Lemma 1 :
includes a lattice isomorphic to L
for every L e S, x e L then every
finite distributive lattice is isomorphic to a lattice in
Proof
Let M be a finite distributive lattice.
:
tion on the size of M.
Otherwise let
z
If
^z}
Let w be the meet of all members
w =
A
W
to N
(u..
>
z.
is a distributive sublattice in which meets and joins are
y = z A u for some u e N.
u.
M has one element the result is immediate.
The minimality property of
preserved.
We argue by induc-
be the smallest member of M which is not the meet of two
members different from z.
H = fy|y
S.
z
implies that if y
Moreover, if u A z = u_ A
1
Aw)
A w) A z = (u~
A z = u«
Aw.
<_
z f z
z then
then
Hence M is isomorphic
By induction hypothesis, N is isomorphic to a lattice in S, so
.
M is isomorphic to a lattice in
Q.E.D.
S.
To complete the proof we will show how to construct a set of
men and women whose preferences yield L
from a set whose preferences
yield L.
Let L be the set of stable matchings possible for women
Lemma_2:
w1 ,...,w
and men m.
m
.
Suppose x =
mnW
(n^W-i
the set of stable matchings for the 2n men nL,...,m
w.
, . . .
,w
m.
;
;
w'
, . . .
,w'
S
n^
;
*"
^ien
m' and women
m!
with the following preferences is isomorphic to L
Use the original preferences of m. in the n-couple situation
for all women strictly preferred to (above)
After w' put w!
w^
Replace w
and finish the ordering arbitrarily.
±
by w|.
:
.
.
-3-
m!
:
First choice w!
11
of m. below w.
w.
Ji=i
:
.
followed by w. and the original preferences
,
Finish arbitrarily.
In the original preference ordering replace m. by (m!,m.)
For m. below m. use (m.,m!).
and all m. above m..
i
3
the original ordering for w
w!
:
i
i
j
Example:
for
if
3
3
is (best) m-num. new ordering is mjm. mlnum.m'
First choice is m. , .
l-l
Second choice is m
.
i
,
followed by m !
i
Finish arbitrarily.
In this definition all arithmetic is modulo n.
We illustrate with
an example after the proof.
Proof
We begin by observing that in any stable 2n-couple matching
:
with these preferences
ferred by m.) must get m
is
the first choice of m!
or m.
m.
,
or m
' .
(3) m.
(4)
.,
hence
m.
gets w'
must get w'
then w!
for all i.
(pre(2) w!
hence w' must get either m._ 1 (and 1 applies),
,
must get somebody at least as good as w!...
does not get w' or w
m! does not get w..
If for some i, m.
(1)
^ +1
»
then m! gets w!
.
(5)
.
If
If m. prefers w. to w',
(Since x is stable w. prefers m. to ml.
would imply m. gets w* or w'
(4)
If m!
got w
so m. and w. would be happier to-
gether.)
These observations imply that nobody gets assigned to the arbitrary
part of his or her ordering.
Further if we are given a stable matching
for the 2n couples we obtain a stable matching for the n-couple problem
(i.e., a member of L) by giving each w. her partner in the 2n-couple
problem, deleting primes where necessary.
Conversely if y € L, there is
a corresponding stable matching for the 2n-couple situation in which m.
is replaced
by m! iff
m.
gets w
or somebody worse in y.
are two 2n-couple matches corresponding to y
— one
If y
<_
x there
in which each m. gets
,
-4-
and one in which each m. gets w!
w'
Those matches in which each m.
...
corresponds to V in the definition of L
gets w!
Example
The four people with preferences given below have stable
:
matching corresponding to a four-element lattice:
w
,
(D)
m
gets w
3
Q.E.D.
.
3
m^ gets w
,
(A)
m.
gets w
,
m„ gets
(abbreviated (2134)) (B) (1243) (C) (1234)
4
(2143).
"l
m
2
m
3
m
W
4
l
W
W
W
2
3
4
(. ..
W
2
W
l
W
l
W
2
f1
W
W
3
4
*3/
W
m
4
W
m
3
2
m
m
m
m
4
2
"l
3
= arbitrary)
3
4
^
is a six-element lattice generated by the preferences:
L
*\ m 2 m 3 m4 "£
W
l
m
2
W
l
W
3
W
4
W
l
w' w' w! w' w
m2
"S
W
W
2
3
w4 w,i ,,i
w 2 ,,t
W „t
w
w
W
ml m2 ml m 3 m4 "H m 2 m 3
4
4
w„ w_ w,
l
m.
w
2
3
l
4
3
m- m, m_ m. m„ m_ m,
m_ nL m' mf mj ml m' m!
w, w_
w' wl
w
m
e • ©
m-f m-f m« m,
• • •
The stable matchings are (213'4'1 2 34)
,
,
,
(1'2'3'4'1234)
(2i3'4«i«2'43), (2'3 4'1'1234), and (2 3'4'1'1243)
,
?
,
,
,
(1 2 3'4'1243)
The last two are
.
members of V.
The construction we have given does not use the smallest number of
people needed to represent a given lattice.
The six-element lattice can
be represented using ten people as follows
^
m
w
w„ w_ w,
2
m
3
m
4
m
w,.
w_ w_
W w_
w. W
w
W W
3
3
W
l
2
5
5
4
w
±
w
2
w
3
m„ m_ m.
"l
m
2
m
m
2
3
w
w,.
4
m,.
m,
m
m
4
5
-5-
The stable matches are (12345),
and (31254).
(12354),
(13245),
(13254),
(31245),
However, it is not possible in general to go from L to L
by adding only one additional couple.
The structure of the set of matches is clearly reminiscent of the
representation of a permutation by cycles.
This theme will be explored
in forthcoming work with Alvin Roth, whose recent work [3] motivated
this note.
-6-
References
1.
Do Gale and L. Shapley, "College Admissions and the Stability of
Marriage," American Math Monthly 69 (1962), pp. 9-15.
2.
D. Knuth,
3.
A.
Marriage Stables , Montreal University Press 1976.
Stability and Incentives,"
Roth, "The Economics of Matching:
to appear in Mathematics of Operations Research.
M/C/294
IECKMAN
t
|NDERV
INC.
JUN95